1. An item is defective (independently of other items) with probability 0.3. You have a method of testing whether the item is defective, but it does not always give you correct answer. If the tested item is defective, the method detects the defect with probability 0.9. If the tested item is good, then the method says it is defective with probability 0.2 a) If an item is tested and the test detects no defect, what is the chance that it is not defective? b) A box contains 4 items. You have tested all of them and the tests detect no defects. What is the probability that none of the 4 items is defective? c) 100 items are tested. Let X = the number of items that are tested defective. Compute EX Compute VarX 2. Roll a single die 10 times. Compute the following probabilities. a) b) c) d) you get at least one 6. You get neither 6 nor 5. You get at least one 6 and at least one 5. You get two 1's three 2's, and five 3's 3. A chocolate egg either contains a toy or is empty. Assume that each egg contains a toy with probability p, independently of other eggs. You have 5 eggs; open the first one and see if it has a toy inside, then do the same for the second one, etc. a) Compute the probability that you get two toys in succession. b) Compute the probability that you get two toys but not in succession. c) Compute the probability that you get at least two toys in succession